Optimization with random and historical vectors

ABSTRACT

The invention relates to the production of an optimum value of a system output function which is dependent upon a plurality of variables where the effect of changes in the variables upon the function may be determined. An optimum is reached quickly and efficiently by changing each variable repetitively in a random manner initially and thereafter the tendency of the changes to be purely random is modified so that past history in the search for the optimum value tends to weight the randomness in favor of the most desirable direction toward optimum.

United States Patent [72] Inventor Robert F. Wheeling Mullica Hill, NJ.[21] Appl. No. 054,503 [22] Filed Sept. 7, 1960 [45] Patented Feb. 16,1971 [73] Assignee Mobil Oil Corporation [54] OPTIMIZATION WITI-l RANDOMAND HISTORICAL VECTORS 8 Claims, 5 Drawing Figs.

[52] U.S.Cl 235/150.l, 235/15 1 12 [51] Int. Cl. G05b 13/00 [50] FieldolSearch 235/151, 184, 185, 193; 23/230,253 (A); 235/151 (E); 235/1 50.1

[56] References Cited UNITED STATES PATENTS 2,980,330 4/1961 Ablow et a1235/184 3,044,701 7/1962 Kerstukas 235/151 39 ADDr---i 37 3,048,3318/1962 Van Nice et al.

2,972,447 2/1961 White 235/151 2,972,446 2/1961 White 235/151 OTHERREFERENCES 1 Eckm-anet al. Optimizing Control ofa Chemical Process?Control Engineering-Sept. 1957. pp. 197 204.

Munson and Rubin, Optimization by Random Search on the Analog Computer.Oct. 25. 1-958; p. 12

Primary Examiner- Eugene G. Botz Attorney- Donald L. Dickerson andOswald G. Hayes ABSTRACT: The invention relates to the production of anoptimum value of a system output function which is dependent upon aplurality of variables where the effect of changes in the variables uponthe function may be determined. An optimum is reached quickly andefficiently by changing each variable repetitively in a random mannerinitially and thereafter the tendency of the changes to be purely randomis modified so that past history in the search for the optimum valuetends to weight the-randomness in favor of the most desirable directiontoward optimum.

1 SO M n NIEU FEB1 SL971 I $564,221

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ROBERT 1-. WHEELING INVENTOR ATTORNEY ROBERT F. WHEELING sum 3 OF 3 FIG.4-.

FIG. 5.

PATENTED FEB 1 6 \sn QBIAS OPTIMIZATION WITH RANDOMAND-I'II STORICALVECTORS This invention-relates to the production of an optimum value ofa function in response to a plurality of dependent variables in whichthe effect of changes of said variables upon said function may bedetermined.

In many systems often there are present a plurality of functions whichmay be selectively controlled or varied independently of one another.Such functions may coact in such systems in an unknown or undefinablemanner to produce a resultant function or functions. Generally, it isdifficult to predict or preset the variable functions to values whichwill produce an optimum resultant function Representative-of such aproblem is that presented in the control ofa chemicalprocessing unit.Materials fed thereto maybe varied over known ranges'both as to rate andquality. Operating condi tions within the unit may also be varied atwill within given ranges. Prior art systems have, by various methods,permitted selection of a combination of the variables so that operationis maintained at a selected point within a permissible. range.

However, optimum operation often is at a point close to a boundary ofpermissible values of controllable parameters. Operations generally maynot be safely maintained at such point so that less advantageous butsafely maintainable sets of operating conditions and material feeds,generally are chosen.

The foregoing is illustrative of the type of operation -in which thepresent invention finds application. The selection of the most desirableoperating point for such a processing unit is a problem of optimization.The optimum may be defined in terms of quality of product produced,quantity for a given input, or may be dependent upon weighing functionsto reflect market prices of the products.

Insofar as the effect of a change in one of the variable functions uponan output function may be determined, an optimum relationship betweenall such variables may be obtained. The process may be understood inconnection with a solution of a problem where a mathematical model orexpression can be precisely formulated. However, a more general problemto which the present invention applies is in relation to complex systemswhere the actual functioning of the systems may not be readily expressedand only the effect on the output of a change in input may be observed.

In accordance with the present invention there is provided a method ofproducing an optimum scalar output from a process unit dependent upon aset of inputs which are variable but which are interrelated and coact toproduce said scalar output. The method comprises generating inputfunctions representative of said inputs and applying said inputfunctions to said process unit for generation of said scalar output. Acondition function is stored which is representative of said scalaroutput. Thereafter, at least said one of said input functions is chargedby an incremental amount of random character to produce a modifiedscalar output and successive changes are thereafter made in at least oneof said inputs by incremental amounts which are of random characterweighted in dependence upon the difference between said scalar outputand modified scalar outputs.

In accordance with a further aspect of the invention, a method ofoptimizing a scalar output function is provided where a set of inputfunctions which are independently variable are interrelated and coact toproduce the scalar output function. The method includes generating inputsignals representative of the input functions and an output signalrepresentative of the scalar output function. An intermediate functionis stored which is representative of the output signal. The inputsignals are then successively changed by incremental amounts of randomcharacter to produce a modified output signal. Thereafter, at least oneof the input signals is changed by an incremental amount, the amountbeing of random character weighted in dependence upon the differencebetween the stored intermediate function and the modified output signal.

In accordance with a further aspect of the invention, there is providedin combination a system having inputs and at least one scalar output,together with a sensing system responsive to variations in said inputsand said output and having means for producing and storing anintermediate function. Means are further provided forprogressivelymodifying the stored intermediate function in response todifferences between the scalar output function and the storedintermediate function upon variations in the inputs. Means are providedfor altering the inputs randomly but successively dependent upon aweighting factor representative of the rate of'change of the scalaroutput function.

For further objects and advantages of the present invention and for amore complete understanding thereof, reference may now be had tothe;following description taken in conjunction with the accompanyingdrawings in which:

FIG. 1 is a plot of time-distance data;

FIG.-2 is a plot of an error function related to FIG. 1;

FIG. 3 illustrates a system for determining an optimum solution for thedata of FIG. 1; FIG. 4 is a modification of the invention in achemicalprocessing system; and 7 FIG. 5 illustrates in schematic formelement 82 of FIGS.

The invention will first be described in connection with a relativelysimple operation in which optimization is employed. The latter operationis in the processing of seismic signals to obtain an indication of thevelocity distribution of earth formations penetrated by such seismicsignals. After presenting the description of the present invention inconnection with this example, there will then be described anapplication of the invention to a more complex problem involving thecontrol of a chemical processing unit.

In seismic exploration where expanding seismic spreads disclosed by Dixin GEOPHYSICS, Vol. XX, pages 68 et seq.,'are employed, it is desireableto process the resultant timedistance data by means of a function of theform:

More particularly, equation (1) is descriptive of line 10 of FIG. 1which is to be fitted to the time-distance data points plotted onFIG. 1. If the points represent time-distance data for a givenreflection and are obtained by employing the above-noted expandingspread techniques and if plotted on scales such that the x scale is interms of distance squared and the y scale is in terms of seismic recordtime squared, then the slope of line 10 is numerically equal to thereciprocal of the square of the average velocity of the earth formationsthrough which the reflection traveled from source thereof to points ofdetection.

In accordance with prior art techniques, the coefficients a and b ofequation (1) have been determined by the process of least-squaresfitting which involves the minimization of:

I). E (21- ill) i=1 where y is the ordinate of line 10 at a givenabscissa, and

y, is the data value at the same abscissa.

The process involved determining the sum of the squares of the valuesAyn Ay uAye. and then adjusting the values of a and b, eithersimultaneously or separately, for successive solutions of equation (2)until values of a and b, which give the error function e of minimumvalue, are determined. The value of the error function is thuscompletely determined by the values of a and b of equation l Bysubstitution of equation (1 in equation (2), the following expression isobtained:

By solving equation (3) for different selected values of a and b, theoptimum values for a and b may be determined;-but only by successiveapproximations. The criterion for such solution is that the summation ofequation (3) be a minimum.

1f equation (3) is solved for a plurality of values of a and b, therewill result data which may be plotted in the form illustrated in FIG. 2.By changing a and b for successive solutions of equation (3), therewould be described a family of ellipses, each-characterized by aconstant value of e, and having a common center. The value of a and bdescribing the common center represents the optimum function forequation (1). The technique for arriving at the optimum function may belikened to mountain climbing, since the data plotted in FIG. 2 may beviewed as a familiar topographic plot of a paraboloid. For data such aswould be plotted in FIG. 2, the desirable end or optimum values for aand b are unknown. By trial and error through solutions of equationssuch as equation (3) an attempt is made to identify the coordinates ofthe center point. The analogy to mountain climbing" is helpful to anunderstanding of the mathematical processes through which successivesolutions to equation (3) lead to the center point.

Prior art methods of optimization have been based upon the philosophythat the best way to determine the direction from a given point 11 tothe center point 12 is systematically to test various values of a and bnear point 11. A solution of equation (3) for each of the points 13, 14,15, and 16 will indicate the directions a and b must be changed in orderto improve the solutions; that is, approach to point 12. By theforegoing method, point 14 obviously would appear to be the bestdirection. Sample solutions would then be obtained for each of points17, 18, and 19. A comparison of the results would then indicate which isthe most desirable direction to go and thusby successive computationsthe entire area represented by FIG. 2 can be sampled to identify thecenter point or optimum solution represented by point 12.

in contrast with the foregoing, the present invention is based upon anovel method of mountain climbing in which the variation froma givendata point as in FIG. 2 is predicated not upon an ordinary sampling ofthe entire area for a solution but upon a random variation in theparameters a and b, to which random variation there is applied ahistorical weighting factor which serves effectively to limit theprobability of following a path in a nonpreferred direction. In the caseillustrated in FIG. 2, the desired direction would be along a straightline connecting points 11 and 12. Since the location of and direction topoint 12 is unknown, a reduction in number of computations to locate thesame is desired. Such reduction is possible through use of the systemillustrated in FIG. 3.

H6. 3 illustrates a system for automatically providing an optimumsolutionto an equation such as equation (3) wherein random variations ofparameters a and b with historical weighting are employed. A solution ofthe relatively simple expression of equation (1) involving but twounknowns is presented primarily by way of example in order to provide anunderstanding of the invention. The invention is deemed to haveprincipal application to systems and methods where a greater number ofvariables are involved. Thus, while the method of the present inventionis wholly operative in systems of fewer variables, it is of principaladvantage where many variable functions are present.

The system of FIG. 3 is designed to provide a solution for equation (3)based upon the availability of the coordinates of the six data pointsplotted in FIG. 1. Equation (3) requires that for the value of x and yat each data point there will be performed a first operation to producethe product ax. The product will then be added to the value of thevariable b from which there will be subtracted the value of y. Theresultant sum or difference is then squared. The sum of all such squaredfunctions represents the error function e.

The system of FIG. 3 includes a multiplier 30, having a first inputconductor connected to a source of voltage 31 representative of thevalue of x,. A second input conductor 32 is provided upon which thevariable a appears as a voltage 13,. The output signal from multiplier30, the product ax is then applied by way of conductor 33 and resistor34 to the input circuit 35 of a squaring unit 36. Resistor 34 forms apart of an adding unit 37. The variable b is applied as a voltage E, tothe adding network 37 by way of conductor 38. A voltage E,, is appliedto the adding network 37 from source 39. Thus. the signal on circuit 35leading to the squaring network 36 is representative of theparenthetical expression of equation (3) for the coordinates x, and y ofthe first data point. This function is then applied to the squaringnetwork 36. The output of the squaring network 36 is then applied to asumming bus 40 by way of adding resistor 41.

identical circuitry is provided for the remaining pairs of data points x,y ...x ,y More particularly, a voltage E from source 50 is applied to amultiplier 51. The conductor 32 is also connected to multiplier 51 forapplication ofa voltage E,,. The output of multiplier 51, a voltage 5;,appearing on conductor 38, and a voltage -E,, are applied to addingnetwork 52, the sum being then applied to the squaring network 53 whoseoutput is applied by way of resistor 54 to the summing bus 40.

A voltage 1'5, & is applied to multiplier 55 along with the voltage E onbu? 32. The output from multiplier 55, the voltage E,, on bus 38, and avoltage --E,, are applied to adding network 56. The resultant sum isapplied to squaring network 57 whose output is applied by way of addingresistor 58 to the summing bus 40.

A voltage E is applied to multiplier 60 along with the voltage E frombus 32. The output of multiplier 60, the voltage E,, on bus 38, and avoltage E, 4 are applied to adding network 61. The resultant sum isapplied to squaring network 62 whose output is applied by way of addingresistor 63 to the summing bus 40. v

A voltage E is applied to multiplier 65 along with the voltage 5,, frombus 32. The output of multiplier 65. the voltage E,, from bus 38, and avoltage E, are applied to adding network 66. The resultant sum isapplied to squaring network 67 whose output is applied by way of addingresistor 68 to the summing bus 40.

Finally, a voltage E, g is applied to multiplier 70 along with thevoltage E from bus 32. The output of multiplier 70, the voltage E,, frombus 38, and a voltage E,, are applied to adding network 71. Theresultant sum is applied to squaring network 72 whose output is appliedby way of adding resistor 73 to the summing bus 40.

The values of x1...x,, and yl...-., are actual data points plotted onFIG. 1 and are constant. In contrast. both the values E,, and E willvary upon operation of the system ultimately to attain a value whichrepresents the optimum solution; i.e., represehtative of point 12 ofFIG. 2. For any given initial set of values for E,, and E there will bedeveloped through the operation of the system thus far described anoutput voltage E, or an error function e on the summing bus 40. Thisfunction will then be stored as a voltage in a storing unit such ascondenser by the momentary closure of switch 81 which is operated underthe control of unit 82 to apply such voltage by way of operationalamplifier A Bus 40 is also con nected directly tocomparison unit 82 at afirst input circuit 84 thereof. Comparison unit 82 also connecteddirectly to condenser 80 at its second input circuit 85. The unit 82 isalso connected by way of linkage 87 to a pair of switches 88 and 89.When switches 88 and 89 are closed. electrical charges are placed onstorage condensers 90 and 91 which are representative of the values aand b, respectively. Such charges are then to be applied to conductors32 and 38, respectively.

The conductor 32 is to be connected to a source of voltage such asbattery by way of a resistor 98 and a normally open switch 99. Battery100 is connected across the terminals of a potentiometer 102. A firstwiper am 103 on the potentiometer 102 is connected to switch 99. Thewiper arm 103 is coupled by driving linkage 104 ton motor 105 cyclicallyto rotate the wiper arm 103. By this means. a voltage is developed onthe wiper arm 103 which varies cyclically from a negative to a positivevalue. The maximum and minimum values of this voltage are dependent uponthe battery 100. A second wiper arm 106 associated with thepotentiometer element 102 is positioned initially at the midpoint of thepotentiometer element 102. Wiper arm 106 is connected to ground. Theupper terminal of condenser 90 may be connected by way of switch 88 andthe operational amplifier A to the upper terminal of a condenser 110 andto the conductor 32. The lower terminals of condensers 90 and 110 areconnected to ground. The wiper arm 106 of potentiometer 102 is connectedby way of driving linkage 111 to a position control unit 112. Thecontrol unit 112 is actuated in response to signals from adifferentiating unit 113, the input of which is connected to the upperterminal of condenser 90.

The upper terminal of condenser 90 may also be connected by way ofswitch 108 and resistor 109 by way of operational amplifier A to theconductor 32. Switches 99 and 108 are interconnected by way of aswitch-actuating linkage 115 leading from a random circuit closing unit116. The random closure unit 116 is also coupled by way ofa switch-camunit 117 to the motor 105. Switch-cam 117 is actuated when arm 103 is atthe beginning of a sweeping cycle on the potentiometer 102. The randomswitch control 116 serves as a time delay for applying an actuatingfunction to switches 99 and 108. The time delay relative to closure ofthe switch-cam vunit 117 may be any fraction of the period of revolutionof arm 103. For successive cycles, the time delay is varied in a randommanner throughout the period. Switch control unit 116 limits closure ofswitches 99 and 108 to one closure for each cycle of revolution of thewiper arm 103. The closure is purposely designed to occur at randomtimes during successive cycles so that there will be applied to theconductor 32 a voltage which is the algebraic sum of the voltage, ifany, at the upper terminal of condenser 90 and the random voltage at thewiper arm 103. That random voltage may be either positive or negativeand of any value between extreme limits determined by the battery 100.

During the first cycle of operations of the system thus far described,the wiper arm 103 will rotate, unit 117 will be actuated, and thereafterswitches 99 and 108 will be closed to establish a voltage on condenser110 which will be maintained constant to provide an initial value of thevoltage E, on conductor 32.

A similar system is provided for production of the voltage E onconductor 38. More particularly, conductor 38 may be connected to asource of voltage such as battery 130 by way of a resistor 131 and aswitch 132. Battery 130 is connected across the terminals of apotentiometer 133. A first wiper arm 134 on the potentiometer 133 isconnected to the switch 132. Wiper arm 134 is coupled to and driven bymotor 105 through linkage 104. By this means, the arm 134 cyclicallysweeps the potentiometer 133. A voltage is thus developed on the wiperarm 134 which varies cyclically from a negative to a positive value. Theextremes of the range of values of this voltage depend upon the battery130. A second wiper arm 135 associated with potentiometer 133 ispositioned normally at the midpoint thereof. Wiper arm 135 is connectedto ground. The upper terminal of condenser 91 may be connected by way ofswitch 89 and operational amplifier A to the upper terminal of acondenser 140 and to the conductor 38. The lower terminals of condensers91 and 140 are connected to ground. The wiper arm 135 of potentiometer133 is connected by way of linkage 136 to a position control unit 141.The control unit 141 is actuated in response to signals from adifferentiating unit 142, the input of which is connected to the upperterminal of condenser 91. The upper terminal of condenser 91 may beconnected by way of switch 145, resistor 146 and operational amplifier Ato conductor 38.

Switches 132 and 145 are interconnected by way of linkage 147 leadingfrom a random circuit closing unit 150. A switchcam unit 151 is actuatedwhen arm 134 is at the beginning ofa sweep cycle on the potentiometer133. The random switch closing unit 150 serves as a time delay followingactuation of switch-cam unit 151 for applying actuating functionsmomentarily to close switches 132 and 145. The time delay will be thattime intervaL between actuation of the switch-cam unit 151 and theactuation of switches 132 and 145. The time delay may thus be anyfraction of the period of revolution of arm 134. For successive cycles,the time delay is varied in a random manner in said period.

While switches 132 and 145 will be closed only once during any givencycle, the closure is purposely designed to occur at random times duringsuccessive cycles so that there will be applied to the conductor 38 avoltage which is the algebraic sum of the voltage, if any, at the upperterminal of condenser 91 and the random voltage at the wiper arm 134.That random voltage may be either positive or negative and of any valuebetween extreme limits determined by the battery 130.

During the first cycle of operations, the wiper arm 134 will rotate,unit 151 will be actuated, and thereafter switches 132 and 145 will beclosed to establish a voltage on condenser 140 which will be maintainedconstant to provide an initial value of voltage E on conductor 38. V

A switch-cam unit 160 is coupled to motor 105 by way of linkage 161.Unit 160 is connected in the control circuit 162 of the comparing unit82. The switch of unit 160 is adapted to be closed at the end of eachcycle of arms 103 and 134. The comparing unit 82 operates to closeswitches 81, 88, and 89 at the end of any given cycle of operationsduring which the value of the voltage applied to comparing condenser 163decreases relative to the value previously stored on the error functioncondenser 80.

Operation of'the system above described for the production of a voltageon conductors 32 and 38, representative of the optimum value of theparameters a and b of equation (3) is as follows. Condensers 80, 90, 91,110, 140, and 163 initially are completely discharged. Source 31 isadjusted to a value representative of an abscissa x,. The source 39 isadjusted to a voltage representative of the ordinate y,. Similaradjustments are made in the sources for E 2 E, and E, 2 ma Motor 105 isthen energized to initiate sweeping action of the arms 103 and 134. Atthe beginning of the first sweep, switch-cam units 117 and 151 apply astart signal to random control units 116 and 150, respectively. Atrandom times during the first cycle of arms 103 and 134, switches 99 and108 momentarily are closed by unit 116. Switches 132 and 145 similarlyare momentarily closed by unit 150. The voltage between tap 103 andground is momentarily applied by way of resistor 98 to conductor 32 andthus causes a charge proportional to that voltage to be stored oncondenser 110.

Momentary closure of switches 132 and 145 apply to conductor 38 avoltage equal to that appearing between tap 134 and ground to store arepresentative charge on condenser 140. The voltages on condensers 110and 140 thus will be the first trial values ofthe a and b parametersofequation (3). Multiplier 30, responsive to the voltage from source 31and the voltage on conductor 32, applies a product voltage to the adder37 by way of conductor 33. The voltage on conductor 38 along with thenegative value of the voltage from source 39 are also applied to adder37. The sum of the voltages applied to the adder 37 are then applied byway of circuit 35 to the squaring unit 36. The output of unit 36 is thenapplied to the summing bus 40. Voltages corresponding to the other fiveterms of the error function can similarly applied to the summing bus 40.Summing bus 40 thus carries a voltage numerically proportional to theerror function 2 corresponding to these trail values of the a and bparameters.

Regardless of the comparing network 82, switch 81 is momentarily closedat the end of the first cycle of operations to store a charge oncondenser representative of the error function. At the same time,switches 88 and 89 are momentarily closed to transfer the chargesoncondensers 110 and 140 respectively to condensers and 91, thusrecording, i.e., storing, the values ofa and b that yielded the valueofthe error function that is stored on condenser 80.

On the next succeeding cycle of operations, the charges on condensersand are altered depending on the random nature of the voltageslfrom arms103 and 134. The altered voltages are then retained on condensers 110and 140 for the production of a new error function on bus 40. The lattererror function appears as a voltage across the comparing condenser 163.At the end of the sweep cycle of arms 103 and 134, the switch-cam unit160 actuates the comparing network 82 so that the voltages acrosscondensers 80 and 163 are compared. Comparing unit 82 is so armed as toactuate switch 81 and switches 88 and 89 only following those operationsof units 160 when the voltage on condenser 163 is less than the voltageon condenser 80. Closures of switches 88 and 89 transfer from condensers110 and 140 respectively to condensers 90 and 91 charges representativeof the voltages on conductors 32 and 38 which improve the solution ofequation (3) Evidence of such improvement is a decrease in voltageacross condenser 163 relative to condenser 80. Closure of switch 81similarly transfers from condenser 163 to condenser 80 a chargerepresentative of the decreased value of the error function.

Following the storage of charges on condensers 90 and 91, the closure ofswitches 99 and 108 by unit 116 on the next succeeding cycle applies toconductor 32 the sum of the voltage on condenser 90 and the voltagesensed by arm 103. Similarly a voltage is applied to conductor 38 whichis representative of the sum of the voltage on condenser 91and thevoltage from arm 134. By this means there will be accumulated oncondensers 90 and 110 charges which will change in the direction of theoptimum value of the parameter a of equation (3). Similarly there willbe accumulated on condensers 91 and 140 charges which will change in thedirection of the optimum value of the parameter b of equation (3Succeeding cycles of operations extended sufficiently in time willdevelop optimum voltages on conductors 32 and 38. This will be so inspite of the fact that any variations in the latter voltage will bedependent upon the random operation of units 116 and 150.

Effectively the system thus far described operates to search part of thearea represented by the plot of FIG. 2 with a random search vector. Theobjective is to move from a given point, i.e., voltages on condensers 90and 91, only if a change is in direction leading toward a minimum valueof the error function, i.e., toward point 12 of FIG. 2.

The present invention contemplates not only searching as above notedwith a random vector but also modifying or weighting the randomcharacter of the search in a direction which is dependent upon theexperience during past cycles of operation. More particularly, thevoltage appearing on condenser 90 is applied by way of conductor 200 tothe dif ferentiating network 113. The output of unit 113 is applied tocontrol unit 112 which by way of linkage 111 serves to move the secondwiper arm 106 and thus alter the probability of selection of positiveand negative values by arm 103. if the change in the voltage oncondenser 90 is rapid as sensed by differentiation, then the arm 106will be moved a substantial distance toward the positive or negative endof potentiometer 102. The magnitude and sense of movement will dependupon the rate and sense of change in voltage on condenser 90. By thismeans, the random vector utilized for searching an optimum will beweighted in favor of those values of search vector which experience hasshown are directed toward optimum. Similar operation is provided for thecontrol of the manner in which the voltage on condenser 91 is modified.The latter voltage is applied by way of conductor 201 to a seconddifferentiating unit 142. Unit 142 actuates the control unit 141 toalter the position of arm 135 to weight the random search vectorcontrolled by unit 150.

By the foregoing means the path followed from a given starting point ofthe values of a and b to an optimum value is made much more direct andrequires many fewer cycles of computation to arrive at the optimumvalue. in short, the comparison circuit 82 through its operation ofswitches 88 and 89 controls the differentiators 113 and 142 so that theposition controls 112 and 141 are effective to move the slide contacts106 and 135 to provide the historical weighting factor which serveseffectively to shorten the search for the optimum value of 5,. Thus thehistorical weighting function is performed only if the voltages oncondensers 90 and 91, upon operation of switches 88 and 89 are changedin directions leading toward a minimum value of the error function 15,.

The foregoing description has been concerned with the solution ofequation (3), the relationships between the parameters of which are wellknown and understood. By the operation of the system thereare producedscalar output voltages onconductors 32 and 38. By applying the inputfunctions x x and y, y to the system, there is generated an intermediatefunction or voltage which is stored on condenser 80. Thereafter thevoltages applied to conductors 32 and 38 are changed by incrementalamounts of random character to modify the scalar output on condenser 80.Successive changes are then made in the voltages on conductors 32 and 38which are random in character but at least in part are weighted independence upon the difference among the successive scalar outputsresulting from such changes.

The foregoing description and the system have been utilized in order toexplain the invention in connection with a relatively simple case. Themore general case has to do with the operation of systems in whichrelationships between the variables and the effects one upon the otherare not known. All that may be available in order to select an optimumset of conditions for controllable parameters are (1) knowledge of thecontrollable parameters at the input, (2) the scalar output from thesystem, and (3) the magnitude and sense of changes in the scalar outputin response to a given change in one or more of the controllable inputs.Those familiar with solution of the problem above illustrated willrecognize that there are ways to arrive at the desired solution whichmay be less cumbersome and perhaps as direct as to use the system ofFIG. 3.

Applicants have found that the use of a random search vector and ahistorical weighting factor is more useful where more than two variablesare involved; though in some two-dimensional cases it is more efficientthan conventional computations. In such multidimensional space, theproblem of search may become astronomical. The present inventionprovides a means for optimization in multidimensional space withincomputational time intervals that renders optimization procedurespractical for control purposes. This will permit operation of a givensystem, for example, at a point near critical value but which point ismore efficient or desirable than could otherwise be maintained.

A representative system is illustrated in FIG. 4. More particularly, thesystem of FIG. 4 is designed to carry out a catalytic cracking processwherein charge stock enters the system by way of channel 220, passingthrough a control valve 221 and a flow telemeter 222. The charge stockthen passes through a heat exchanger 223 and thence to a heater 224. Theoutput of heater 224 passes through flow path 225 to a catalytic processvessel or case 226. The output flow path 227 from the case 226 leads tothe heat exchange unit 223 and thence to a fractionating tower 228.Light fractions pass from the fractionating tower 228 by way ofcondenser 229 to a surge tank 230. Noncondensible components pass by wayof telemeter 231 to a storage unit 232. Condensate from tank 230 passesto a proportioning valve 239. Part of the condensate is forced by pump240 through a telemeter 241 into the fractionating tower 228 near thetop thereof. The remainder of the condensate passes through a telemeter245 andthencc to a storage unit 246. An intermediate fraction of theproduct from the fractionating tower 228 passes through telemeter 250and thence to a storage unit 251. The heaviest components from thefractionating tower 228 pass to a proportioning valve 255. A portion ofsuch heavy components is fed by way of telemeter 256 back into thecharge stock line 220 for recycling. The remainder of the heavycomponents passes by way oftelemeter 257 to a storage unit 258.

While the processes performed in the system thus far described are wellknown, it is to be understood that various conditions of feed rate,temperature and pressure, reflux ratio, and recycle ratio may be varied.For example, the temperature to which the charge stock is heated infurnace 224 may be varied and controlled by unit 260. The temperatureand rate of flow of an eutectic solution passing through the case 226may be controlled by unit 261. The temperature maintained in thefractionating tower 228 may be varied and controlled by unit 262. Thus,there may be produced in response to operation of this system controlledflow to each of the storage units 232, 246, 251, and 258. Theproportions of the charge stock which ultimately are processed and reacha given destination will be determined largely by the conditionsmaintained in the system.

In accordance with the present invention, the parameters of the systemat various critical points are sensed and applied to a computer 270. Theoutput of the computer is then utilized to modify the controllableconditions in the processing system so that the products resultingtherefrom will be of optimum character. The computer 270 in itspreferred form will comprise a digital computer capable of carrying outfunctions of the type above explained in connection with FIGS. 1-3. Itscapability is such that it may operate on the multiplicity of variablespresent in the system of FIG. 4. An analogue system would be capable ofso operating, but the capacity of a digital unit is such that it lendsitself more readily to problems of this nature than an analogue system.For this reason, the representation of heaviest digital computer inblock form his been adopted for the purpose of the present description.

Input data are provided for computer 270 by means of the varioustelemeters. More particularly, the input flow rate under the control ofvalve 221 is sensed by the telemeter 222 and applied to the computer byway of the signal channel 271 (shown dotted). The temperaturerepresentative of the operation of heater 224 is applied as input databy way of channel 272. The temperature representative of that maintainedby the eutectic solution in case 226 is applied as input data by way ofchannel 273. The reflux ratio of the products from condenser 230 isapplied as data by way of channel 274 leading to transducer 241 andchannel 275 leading to transducer 245. A signal from control unit 262representing the temperature in the fractioning tower 228 is applied asinput data by way of channel 276. A signal representing light endproducts flowing to storage 232 are applied as input data by way ofchannel 277 leading to telemeter 231. A signal representative of theintermediate fraction flowing to storage 251 is applied as input data byway of channel 278 which leads to telemeter 250. The recycle ratio ofthe heavy fraction flowing from fractionating tower 228 is applied asinput data by way of a signal on channel 280 leading from telemeter 257and a signal on channel 281 leading from telemeter 256.

Control linkages are provided between the computer 270 andvariouscontrol elements in the system. More particularly, the computermay control operation of the heater unit 260 by way of linkage 300.Control of the eutectic solution may be effected by way of linkage 301.The temperature of the fractioning tower 228 is controlled by way oflinkage 303. The reflux ratio from tank 230 is controlled by way oflinkage 305 leading to proportioning valve 239. The ratio of recycle ofheavy ends from fractionating tower 228 is controlled by way of linkage304 leading to valve 255. Finally, the flow rate of the charge stock iscontrolled by linkage 302 leading to the valve 221.

With the system operating under the control of computer 270,considerable flexibility is available in establishing optimum operation.The volume of products flowing to any one of storage units 232, 246,251, or 258 may be required to be maximum. Alternatively, the productsmay be proportioned in dependence upon market conditions so that thevalue of the combined products from the system will be a maximum. Suchcriteria may be established in the computer 270 on the same generalbasis as the minimization criteria of FIG. 3 and conditions within thesystem will be automatically adjusted for such optimum operating point.

In FIG. 4 a complete mathematical model of the system is not availableand is not necessary. All that is necessary is that the values of thevarious output flow rates be monitored by the computer and that thecomputer contain an optimizer embodying the principles of thisinvention. The computer 270 may then search the various controllableconditions with a random vector as was done in the case of FIG. 3 andthen apply historical weighting to such random vector which by means oflinkages 300-305 produce stepwise changes in the various parameters.Ultimately, an optimum operating condition of the system is attained.Thus, computer 270 is functionally representative of the embodimentshown in FIG. 3. except E E ind E, as well as a and b. The above exampleillustrates that, insofar as it may be feasible to make measurementsdirectly on the physical system under consideration, mathematical modelsmay be unnecessary. The invention is thus applicable to optimization ofsystems mathematical, physical or combinations, provided that the outputto be optimized is generated responsive to a plurality of inputs.

Through the use of applicants optimization procedure. the presentinvention permits the use of direct control of the process unit eventhough many variables are involved since the computation time toestablish optimum may be so reduced as to be compatible with periodsduring which control action must be taken in the actual process unit.

The processing unit illustrated in FIG. 4 is diagrammaticallyrepresentative of a catalytic cracking unit in which the case 226 alongwith cases 2260 and 226b may be sequentially connected between flowlines 225 and 227 and to a source of air 226c to permit periodicregeneration of a catalyst in each processing case while maintaining acontinuous flow through the system. It may be found desirable to controlparameters or variables in the system other than those specificallydescribed above. However, the foregoing is to be taken as representativeof applicants method of optimizing wherein actual control of afunctional unit may be provided.

The system of FIG. 4 has been illustrated to emphasize functionalrelationships therein. It will be appreciated that the number of actualinput functions applied to computer 270 is greater than necessarybecause the computer itself may include sensing devices so that itknows" the actual settings of control points. For example, the computer,having set valve 221, already is in possession of data representative ofsuch settings. This being the case, the data input circuit 271 may beeliminated because its function may be carried out by means of aninternal connection or function of the computer 270. Other points in thesystem may have a similar relation with respect to the computer 270. i

It will now be appreciated that the invention may be carried out by handeven though by so doing the primary advantage of following applicantsinvention would be greatly reduced. In manual operation of the system ofFIG. 3, for example, the arms 103 and 134 are adjusted manuallyto'random locations on the potentiometers. Momentary closures ofswitches 99, 108, 132, and 145 are manually accomplished so that anerror function is produced across condenser 163. Manual closure ofswitch 81 transfers a charge to condenser 80. Thereafter, subsequentcycles of random potentiometer settings and subsequent switchingoperations manually performed produce comparison error functions oncondenser 163. Switches 88 and 89 are then manually closed only when theerror function is reduced relative to a preceeding cycle of operation.The potentiometer arms 106 and 135 are then manually moved in suchdirection as to increase the probability of randomly selecting voltagesfrom batteries and for application to conductors 32 and 38. Ultimately,optimum values will appear on the latter conductors.

Having described the invention in connection with the analogue device ofFIG. 3 and the computer-process system of FIG. 4, it will be appreciatedthat many of the components of each system have not'been described indetail because both functionally and structurally they are well known tothose skilled in the art. The telemeters such as telemeter 22 of FIG. 4may be of any one of various systems which generate an electrical orpneumatic signal proportional to and dependent upon the rate of flow offluid therethrough.

In FIG. 3, differentiation as carriedout by unit 113 is well known tothose skilled in the art. The position control unit 112 may comprise aservomechanism for positioning the arm 106. The comparison circuit 82may be of the type known as a cathode-coupled, binary comparator such asillustrated at page 169 of PULSE AND DIGITAL CIRCUITS, Millman and Taub,McGraw Hill, 1956. A circuit of the latter type is illustrated in FIG.wherein the condenser 163 is connected by way of conductor 84 to theinput grid of a tube 400. The voltage on condenser 163 will change inresponse to changes in voltage on the summing bus 40. Condenser 80 isconnected by way of conductor 85 to the control grid circuit of tube401. The output of tube 401 is coupled by way of a blockingoscillator-amplifier unit 402 to a relay coil 403. Switch 81 is actuatedby relay coil 403. A circuit including tubes 400 and 401 operates toproduce an output signal when the voltage across condenser 163 exceedsthe voltage across condenser 80. The latter signal is amplified and apulse is produced in unit 402 and applied to relay 403 to close acircuit through switch 81 for analyzing the voltages on condensers 163and 80. Linkage 87 may then extend to switches 88'and 89, FIG. 3, totransfer to condensers 90 and 91 respectively the tentative incrementvalues stored on condensers 110 and 140 respectively. Such transfer willtake place only when the voltage on condenser 163 changes relative tothe voltage on condenser 80 in one of two senses. The voltage oncondenser 80 may be made optimum, either a maximum or a minimum, inaccordance with the present invention, depending upon polarities of thevoltages employed and the manner in which condensers 163 and 80 areconnected into the circuit of FIG. 3. In accordance with the operationof the system of FIG. 3, it is desired to so arrange the circuit thatthe error voltage appearing on conductor 40 will be minimized. Incontrast, in the system as illustrated in FIG. 4, it may be desirable toproduce output quantities or products of maximum character.

Since the motor 105 of FIG. 3 drives the potentiometer arms 103 and 134cyclically, it is desirable to alter the condition of charge stored oncondensers 163 and 80 only at the end of each cycle of operations of thepotentiometers. For this purpose, the cam-operated switch 160 has beenprovided. Switch 160 is connected to the comparison circuit 82 by way ofconductor 162 which is shown in FIG. 5 as leading to one terminal of asecond relay 410. The second terminal of relay 410 is connected by wayof battery 411 to ground. By this means, when the cam-operated switch160 is closed, relay 410 will be energized to close the switch 412 whichis connected in series with switch 81. Thus, equalization of theconditions on condensers 80 and 163 will be carried out periodically butonly if the charge on condenser 80 differs from the charge on condenser163 in one sense only positive or negative, but not both. By this means,a system is provided having at least two inputs into which scalar inputquantities, either electrical signals as in FIG. 1 or materials to beprocessed in FIG. 4, are fed. The system has at least one output fromwhich a resultant scalar output, the charge on condenser 163, FIG. 1, orproducts from the system of FIG. 4, is obtained. As to the two inputquantities, a first control means is provided for changing from aninitial level to a different level a characteristic of the quantityapplied to the first of the inputs. Similarly, a second control means isprovided for changing from an initial level to a different level acharacteristic of the quantity applied to the second of the inputs. Afirst change selector is provided for ac tuating the first control meansin a random fashion both as to sense and magnitude. A second changeselector is provided for periodically actuating the second control meansboth as to sense and magnitude. Storage of a first conditionrepresentative of the initial level of the output quantity and storageof a second condition representative of the level of the output quantityfollowing each actuation of the control means are then effected. Thestored conditions are then compared and means responsive to differencesbetween the first and second conditions, of one sense only, are providedfor establishing a new initial level for each of the input quantitiesand for the output quantity where the changes in the input quantitiesare equal to the changes made by the first and second change selectorsrespectively and where the new initial level for the first storedcondition corresponds with the second stored condition.

In a further aspect, applicantshave provided a method in which there aregenerated input functions representative of independent variables ateach of a series of states of an operating system. In response to suchvariable functions, there is produced a signal representative of thevalue of the output function from the system. The latter signal is thencompared for each of a plurality of states of the independent variablesby progressively varying the input functions representative ofsuccessive states conformably with the following schedule:

A. The variation between the initial state and a second state isrepresentable by a random vector, and

B. The variation between a given state. subsequent to the second stateand the state immediately precedent thereto is the weighted resultant ofa random vector and a controllable vector representative of variationsprior to the immediately precedent state and weighted to a degreerepresentative of the quantity by which earlier variations shall haveresulted in successive states wherein comparison shall have demonstratedchanges of said output function in the direction of a desired optimum.

Having described the invention in'connection with certain modificationsthereof, it is to be understood that further modifications may nowsuggest themselves to those skilled in the art and it is intended tocover such modifications as fall within the scope of the appendedclaims.

lclaim:

1. ln optimizing a scalar output function of a system where a set ofinput values to said system some of which are independently variable,are interrelated and coact to produce said scalar output function; thecombination of:

means for generating input values including those representative of saidvariables at each of a series of states of said system;

means for calculating from said input values the value of said scalarfunction at each such state;

means for storing said value of each said state;

means for comparing the value of said scalar function as calculated ateach said state with the value thereof as calculated at a precedentstate;

means for randomly varying said independently variable input values; and

means for weighting the random variations of said independently variableinput values by amounts related to the magnitude of the earliervariations which produced change of said scalar function toward thedesired optimum.

2. In optimizing a scalar output function ofa system where a set ofinput values to said system some of which are independently variable,are interrelated and coact to produce said scalar output function, themethod which comprises:

generating in said system input signals representative of said inputvalues, some of which are independently variable corresponding with saidindependently variable input values;

generating from said input signals an output signal representative ofsaid scalar output function;

storing in said system an intermediate function representa tive of saidoutput signal;

repetitively changing said variable input signals by amounts of randomcharacter to produce output signals of differing magnitude some of whichrepresent improved scalar output functions; and

in response to said changes of said variable input signals whichproduced said improved output functions, changing said variable inputsignals by an amount of random selection which selection is weighted independence upon the earlier changes in the values of said variable inputsignals which produced said improved output functions.

3. In optimizing a scalar output function ofa system where a set ofinput values to said system some of which are independently variable,are interrelated, and coact to produce said scalar output function, themethod which comprises:

generating in said system input signals representative of said inputvalues, some of which are independently variable corresponding with saidindependently variable input values;

generating from said input signals an output signal representative ofsaid scalar output function; storing in said system an intermediatefunction representative of said output signal;

periodically changing said variable input signals by amounts of randomcharacter to produce output signals of differing magnitude some of whichrepresent improved scalar output functions; and

thereafter weighting the random change of said variable input signals byamounts related to the magnitude of the earlier changes which producedsaid improved output functions.

4. in optimizing a scalar output function of a system where a set ofinput values to said system some of which are indepen dently variable,are interrelated, and coact to produce said scalar output function, themethod which comprises the following steps:

generating in the system input signals representative of said inputvalues, some of which are independently variable corresponding with saidindependently variable input values; generating from said input signalsan output signal representative of said scalar output function; storingin said system an intermediate tive of said output signal; changing saidvariable input signals by amounts of random character to produce outputsignals of differing magnitude some of which may produce output signalsrepresentative of improved scalar output functions; thereafterrepetitively changing said variable input signals by amounts dependentupon random selections; and weighting the random selections by amountsrelated to the magnitude of the earlier changes which produced outputsignals representing improved scalar output functions.

5. The combination which comprises a process unit in which there isdeveloped a physical output quantity which is to be optimized and whichunit has a plurality of control elements, control means connected tosaid elements for changing control characteristics of said elements frominitial levels to incrementally different levels, change selectormeans'for each said control means for repeatedly actuating the samerandomly both as to sense and magnitude, means for storing a firstphysical quantity responsive to and representative of a scalarcharacteristic of said output quantity, means for generating a secondphysical quantity responsive to and representative of saidcharacteristic of said output quantity following each change in saidelements, comparison means responsive to differences in magnitudebetween said first physical quantity and said second physical quantityof one sense only for establishing new initial levels in said controlcharacteristics and for changing the storage of said first physicalquantity to a level corresponding with that of said second physicalquantity, and means for interconnecting said control elements and saidchange selector means for modifying the random character of said changeselector means in accordance with the rates of changes of said controlelements from the respective initial levels thereof to new levels.

6. The combination which comprises a process unit in which there isdeveloped a physical output quantity which is to be optimized and whichunit has a plurality of control elements, control means connected tosaid elements for changing control characteristics of said elements frominitial levels to incrementally different levels, change selector meansfor each said control means for repeatedly actuating the same randomlyboth as to sense and magnitude, means for storing a first physifunctionrepresentacal quantity responsive to and representative of a scalarcharacteristic of said output quantity, means for generating a secondphysical quantity responsive to and representative of saidcharacteristic of said output quantity following each change in saidelements, comparison means responsive to differences in magnitudebetween said first physical quantit and said second physical quantity ofone sense only for esta lShing new initial levels in said controlcharacteristics and for changing the storage of said first physicalquantity to a level corresponding with that of said second physicalquantity, means for separately interconnecting each said control elementand its associated change selector means for modifying the randomcharacter of each said change selector means in accordance with the rateof change of said control elements from the respective initial levelsthereof to new levels.

7. A system of optimizing the magnitude of a scalar output function of asystem where a set of input values to said system some of which arevariable, are interrelated and coact to produce said scalar outputfunction, which comprises:

.means for generating in said system input signals representative ofsaid input values, some of which are independently variablecorresponding with said independently variable input values;

means for generating from said input signals an output signalrepresentative of the magnitude of said scalar output function;

means for repeatedly and randomly varying said variable input signalsfor producing a plurality of values of said output signal some of whichrepresent change in said scalar output function in the optimaldirection;

means for storing said output signals representing said change in saidoutput function in said optimal direction;

comparison means for comparing each new value of said output signal withthe stored value of the output signal;

means for weighting subsequent random changes of said variable inputsignals by amounts which increase with the extent to which previouschanges of said variable input signals caused said output function toapproach its 0ptimum value; and

means operative under the control of said comparison means forcontrolling said weighting means for operation only when said new valuesof said output signal represent change of said output function towardits optimum value. 8. in optimizing the magnitude of a scalar outputfunction of a system where a set of input values to said system some ofwhich are variable, are interrelated and coact to produce said scalaroutput function, the method which comprises:

generating in said system input signals representative of said inputvalues, some of which are independently variable corresponding with saidindependently variable input values; generating from said input signalsan output signal representative of the magnitude of said scalar outputfunction;

repeatedly and randomly varying said variable input signals forproducing a plurality of values of said output signal some of whichrepresent change in said scalar output function in the optimaldirection;

storing said output signals representing said change in said outputfunction in said optimal direction; and

in response to said random variations of said variable input signalswhich produced change of said scalar function in said optimal directionweighting subsequent random changes of said variable input signals byamounts which increase with the extent to which previous changes of saidvariable input signals caused said output function to change in saidoptimal direction.

1. In optimizing a scalar output function of a system where a set ofinput values to said system some of which are independently variable,are interrelated and coact to produce said scalar output function; thecombination of: means for generating input values including thoserepresentative of said variables at each of a series of states of saidsystem; means for calculating from said input values the value of saidscalar function at each such state; means for storing said value of eachsaid state; means for comparing the value of said scalar function ascalculated at each said state with the value thereof as calculated at aprecedent state; means for randomly varying said independently variableinput values; and means for weighting the random variations of saidindependEntly variable input values by amounts related to the magnitudeof the earlier variations which produced change of said scalar functiontoward the desired optimum.
 2. In optimizing a scalar output function ofa system where a set of input values to said system some of which areindependently variable, are interrelated and coact to produce saidscalar output function, the method which comprises: generating in saidsystem input signals representative of said input values, some of whichare independently variable corresponding with said independentlyvariable input values; generating from said input signals an outputsignal representative of said scalar output function; storing in saidsystem an intermediate function representative of said output signal;repetitively changing said variable input signals by amounts of randomcharacter to produce output signals of differing magnitude some of whichrepresent improved scalar output functions; and in response to saidchanges of said variable input signals which produced said improvedoutput functions, changing said variable input signals by an amount ofrandom selection which selection is weighted in dependence upon theearlier changes in the values of said variable input signals whichproduced said improved output functions.
 3. In optimizing a scalaroutput function of a system where a set of input values to said systemsome of which are independently variable, are interrelated, and coact toproduce said scalar output function, the method which comprises:generating in said system input signals representative of said inputvalues, some of which are independently variable corresponding with saidindependently variable input values; generating from said input signalsan output signal representative of said scalar output function; storingin said system an intermediate function representative of said outputsignal; periodically changing said variable input signals by amounts ofrandom character to produce output signals of differing magnitude someof which represent improved scalar output functions; and thereafterweighting the random change of said variable input signals by amountsrelated to the magnitude of the earlier changes which produced saidimproved output functions.
 4. In optimizing a scalar output function ofa system where a set of input values to said system some of which areindependently variable, are interrelated, and coact to produce saidscalar output function, the method which comprises the following steps:generating in the system input signals representative of said inputvalues, some of which are independently variable corresponding with saidindependently variable input values; generating from said input signalsan output signal representative of said scalar output function; storingin said system an intermediate function representative of said outputsignal; changing said variable input signals by amounts of randomcharacter to produce output signals of differing magnitude some of whichmay produce output signals representative of improved scalar outputfunctions; thereafter repetitively changing said variable input signalsby amounts dependent upon random selections; and weighting the randomselections by amounts related to the magnitude of the earlier changeswhich produced output signals representing improved scalar outputfunctions.
 5. The combination which comprises a process unit in whichthere is developed a physical output quantity which is to be optimizedand which unit has a plurality of control elements, control meansconnected to said elements for changing control characteristics of saidelements from initial levels to incrementally different levels, changeselector means for each said control means for repeatedly actuating thesame randomly both as to sense and magnitude, means for storing a firstphysical quantity responsive to and representative of a scalarcharacteristic of said output quantity, meAns for generating a secondphysical quantity responsive to and representative of saidcharacteristic of said output quantity following each change in saidelements, comparison means responsive to differences in magnitudebetween said first physical quantity and said second physical quantityof one sense only for establishing new initial levels in said controlcharacteristics and for changing the storage of said first physicalquantity to a level corresponding with that of said second physicalquantity, and means for interconnecting said control elements and saidchange selector means for modifying the random character of said changeselector means in accordance with the rates of changes of said controlelements from the respective initial levels thereof to new levels. 6.The combination which comprises a process unit in which there isdeveloped a physical output quantity which is to be optimized and whichunit has a plurality of control elements, control means connected tosaid elements for changing control characteristics of said elements frominitial levels to incrementally different levels, change selector meansfor each said control means for repeatedly actuating the same randomlyboth as to sense and magnitude, means for storing a first physicalquantity responsive to and representative of a scalar characteristic ofsaid output quantity, means for generating a second physical quantityresponsive to and representative of said characteristic of said outputquantity following each change in said elements, comparison meansresponsive to differences in magnitude between said first physicalquantity and said second physical quantity of one sense only forestablishing new initial levels in said control characteristics and forchanging the storage of said first physical quantity to a levelcorresponding with that of said second physical quantity, means forseparately interconnecting each said control element and its associatedchange selector means for modifying the random character of each saidchange selector means in accordance with the rate of change of saidcontrol elements from the respective initial levels thereof to newlevels.
 7. A system of optimizing the magnitude of a scalar outputfunction of a system where a set of input values to said system some ofwhich are variable, are interrelated and coact to produce said scalaroutput function, which comprises: means for generating in said systeminput signals representative of said input values, some of which areindependently variable corresponding with said independently variableinput values; means for generating from said input signals an outputsignal representative of the magnitude of said scalar output function;means for repeatedly and randomly varying said variable input signalsfor producing a plurality of values of said output signal some of whichrepresent change in said scalar output function in the optimaldirection; means for storing said output signals representing saidchange in said output function in said optimal direction; comparisonmeans for comparing each new value of said output signal with the storedvalue of the output signal; means for weighting subsequent randomchanges of said variable input signals by amounts which increase withthe extent to which previous changes of said variable input signalscaused said output function to approach its optimum value; and meansoperative under the control of said comparison means for controllingsaid weighting means for operation only when said new values of saidoutput signal represent change of said output function toward itsoptimum value.
 8. In optimizing the magnitude of a scalar outputfunction of a system where a set of input values to said system some ofwhich are variable, are interrelated and coact to produce said scalaroutput function, the method which comprises: generating in said systeminput signals representative of said input values, some of which areindependently variable corresponding witH said independently variableinput values; generating from said input signals an output signalrepresentative of the magnitude of said scalar output function;repeatedly and randomly varying said variable input signals forproducing a plurality of values of said output signal some of whichrepresent change in said scalar output function in the optimaldirection; storing said output signals representing said change in saidoutput function in said optimal direction; and in response to saidrandom variations of said variable input signals which produced changeof said scalar function in said optimal direction weighting subsequentrandom changes of said variable input signals by amounts which increasewith the extent to which previous changes of said variable input signalscaused said output function to change in said optimal direction.